# Laplace Transform of Real Power

## Theorem

Let $n$ be a constant real number such that $n > -1$

Let $f: \R \to \R$ be the real function defined as:

$\map f t = t^n$

Then $f$ has a Laplace transform given by:

 $\displaystyle \laptrans {\map f t}$ $=$ $\displaystyle \int_0^\infty e^{-s t} t^n \rd t$ $\displaystyle$ $=$ $\displaystyle \frac {\map \Gamma {n + 1} } {s^{n + 1} }$

where $\Gamma$ denotes the gamma function.

## Proof

 $\displaystyle \laptrans {t^n}$ $=$ $\displaystyle \int_0^\infty e^{-s t} t^n \rd t$ Definition of Laplace Transform $\displaystyle$ $=$ $\displaystyle \int_0^\infty e^{-u} \paren {\dfrac u s}^n \rd \paren {\dfrac u s}$ Integration by Substitution: $u := s t$ where $s > 0$ is assumed $\displaystyle$ $=$ $\displaystyle \dfrac 1 {s^{n + 1} } \int_0^\infty u^n e^{-u} \rd u$ $\displaystyle$ $=$ $\displaystyle \dfrac {\map \Gamma {n + 1} } {s^{n + 1} }$ Definition of Gamma Function

$\blacksquare$